Recitation 18
BJT: Regions of Operation & Small Signal Model
6.012 Spring 2009
Recitation 18: BJT-Regions of Operation & Small
Signal Model
BJT: Regions of Operation
System of equations that describes BJT operation:
qVBE
qVBC
Is
qVBC
) − exp(
) −
(exp(
) − 1)
Ic = Is exp(
kT
kT
βR
kT
Is
qVBE
Is
qVBC
IB =
(exp(
) − 1) +
(exp(
) − 1)
kT
kT
βF
βR
qVBE
qVBC
Is
qVBE
IE = −Is (exp(
) − exp(
)) −
(exp(
) − 1)
kT
kT
kT
βF
Is =
βF =
βR =
qAE n2i DnB
NaB wB
NdF DnB wE
NaB DpE wB
NdC DnB wC
NaB DpC wB
This set of equations can describe all four regimes of operation for BJT
Forward Active: VBE > 0, VBC < 0
1
Recitation 18
BJT: Regions of Operation & Small Signal Model
Reverse Active (RAR)
VBE < 0, VBC > 0
Cut-off
VBE < 0, VBC < 0
Saturation
VBE > 0, VBC > 0
2
6.012 Spring 2009
Recitation 18
BJT: Regions of Operation & Small Signal Model
6.012 Spring 2009
Understanding the IC vs. VCE curve: IC drops rapidly below VCE,SAT 0.1 to 0.2 V.
Why?
• Each curve IB is fixed
• VCE = VBE − VBC , =⇒ VBC = VBE − VCE
• When VCE is large, VBC < 0, FAR. As we reduce VCE , VBC reduces, at some point, VBC
starts to become forward biased. Now, hole flux from B → C increases exponentially
from Law of Junction; to keep IB constant, hole flux into emitter must be reduced,
=⇒ VBE drops, =⇒ IC drops quickly.
Small Signal Model of BJT
(Next week we will be using BJT & MOSFET for amplifier circuits) Want to know the
small signal circuit model of BJT
δic 1. Transconductance gm =
δVBE Q
Ic = Is eqVBE /kT =⇒ gm =
Note, different from MOSFET: gm q
Ic
Is eqVBE /kT =
kT
Vth
w
2 ID (depends upon device size), but not for
L
bipolar case.
3
Recitation 18
BJT: Regions of Operation & Small Signal Model
6.012 Spring 2009
2. Input resistance:
IB =
gπ =
or γπ =
Is qVBE /kT
e
βF
1
δiB
IB
gm
=
=
=
γπ
δVBE
Vth
βF
βF
gm
The input resistance of MOSFET is ∞. In order to have a high input resistance for
BJT, need high current gain βF
Example: npn with βF = 150, Ic = mA
gm =
gπ =
Ic
1 × 10−3 A
= 40 mS
=
Vth
0.025 V
40 mS
gm
=
= 0.267 mS (γπ = 3.7 kΩ)
βF
150
3. Output resistance: Ebers-Moll model have perfect current source in FAR. Real characteristics show some increase in ic with VCE
δic
where does go come from?
δVCE
qAE n2i DnB qVBE /kT
= Is eqVBE /kT =
e
NaB wB
go =
In FAR: Ic
wB shrinks as |VBC | ↑, thus Ic ↑.
Model: go = slope =
go =
1
Ic
=
γo
VA
Ic
Ic
(VA VCE )
VCE + VA
VA
Example: Ic = 100 μA, VA = 35 V, =⇒ γo = 350 kΩ
VA increases with increasing base width and increasing base doping. This is also why NaB
usually NdC
4
Recitation 18
BJT: Regions of Operation & Small Signal Model
6.012 Spring 2009
Now what do we have so far? Need to add capacitances...
Junction Capacitance (depletion capacitance)
qs NaB NdE
(∵ NdE NaB )
2(NaB + NdE )(φBE − VBE )
qs
NdC
qs NaB NdC
=
2(NaB + NdC )(φBC − VBC )
2 (φBC − VBE )
(B-E): CjE =
(B-E): CjC
• Both are functions of bias
• Since NaB NdC , CjE CjC . CjC is often called Cμ .
Diffusion Capacitance
Cb =
|QnB | =
=
Cb =
2
wB
2Dn
δ
|QnB |
δVBE
1
qAE wB npBO eqVBE /kT
2
2 wB
wB
qAE DnB
1
qVBE /kT
wB
npBO e
=
Ic
2
DnB
wB
2DnB
2 2 wB
wB
δ
ic =
gm
δVBE
2DnB
2Dn
= τF base diffusion transit time
5
Recitation 18
BJT: Regions of Operation & Small Signal Model
Cb is in between base and emitter:
Cb + CjE = Cπ
Add the following
• depletion capacitance: collector to bulk CCS
• parasitic resistances: γb of base, γex of emitter, γc of collector
Complete small signal model
6
6.012 Spring 2009
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